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Introduction

The density function of the inverse-Wishart distribution:

$$ f(\mathbf{X}; \boldsymbol{\Psi}, \nu) = \dfrac{|\boldsymbol{\Psi}|^{\frac{\nu}{2}}}{2^{\frac{\nu p}{2}} \Gamma_p\left( \frac{\nu}{2} \right)} | \mathbf{X} |^{-\frac{\nu + p + 1}{2}} \exp \left( - \frac{1}{2} \text{tr} (\boldsymbol{\Psi} \mathbf{X}^{-1}) \right) $$

where $\Gamma_p$ is the multivariate gamma function.

Parameters:

• Psi_par is $\boldsymbol{\Psi}$
• nu_par is $\nu$

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Density

Definition:

template<typename mT, typename pT>
statslib_inline
return_t<pT> dinvwish(const mT& X, const mT& Psi_par, const pT nu_par, bool log_form = false);

Computes the density function.

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Examples:

// parameters
arma::mat Psi = arma::eye(5,5);
double nu = 10.0;

// Armadillo input
arma::mat X(10,1);
X.fill(2.5);

arma::mat dens_vals_mat = stats::dinvwish(X,Psi,nu);
arma::mat log_dens_vals_mat = stats::dinvwish(X,Psi,nu,true);

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Random Sampling

Definition:

 
template<typename mT, typename pT>
statslib_inline
mT rinvwish(const mT& Psi_par, const pT nu_par, const bool pre_chol = false);

Generates pseudo-random draws.

Here pre_chol indicates whether Psi_par is being passed as a Cholesky decomposition (in lower triangular format).

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Examples:

// parameters
arma::mat Psi = arma::eye(5,5);
double nu = 10.0;

// Armadillo output
arma::mat rand_mat = stats::rinvwish<arma::mat>(Psi,nu);

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